An introduction to the modern theory of equations, by Florian Cajori.

BINOMIAL AND I i:tE(C.PROCAL EQUATIONS

81

V. The solution of x' 1 -1 = 0, where n is any composite number,
is reduced to the solution of binomial equations in which n is a
prime number.
This important result, of which further use will be made in
a later chapter, follows readily from the theorems III and IV
of this paragraph.
67. Depression of Reciprocal Equations. A reciprocal equation
of the standard form (~ 31) can always be depressed to one of half
the dimensions.
Divide both sides of the given reciprocal equation
aOx'2m + am- + C2 "- + a' x + a o 0
by xm, and we get, on collecting in pairs the terms which are
equidistant from the beginning and end,
ao(m+ _ al(m + + ia1)+.+ am, -) = -O
Assuming y = x +, we obtain
XI + = (2 + +{ ) -Y =3 -3 y,
4 +14 (X3 + )( + 2 +2 =y4-4 y2+2,
x4 + x3 x
and generally
x +- 1+) I =\- 1+ )
By substitution in the above equation we obtain an equation of the mth degree in y. From the relation x + - = y we
see h t,. u x
see that two values of x may be deduced from each value of y.
G